1. Raga Gopalakrishnan University of Colorado at Boulder Adam Wierman (Caltech) Amy R. Ward (USC) Sherwin Doroudi (CMU) Staffing and Routing to incentivize servers in many-server systems

2. strategic servers system performance Journal reviews Call centers Crowdsourcing Cloud computing Enterprise data centers … service systems 

3. strategic servers system performance Journal reviews Call centers Crowdsourcing Cloud computing Enterprise data centers … service systems This talk: Impact of strategic servers on optimal system design  Classic Queueing: Assumes fixed (arrival and) service rates, fixed control/policies. Queueing games: Strategic arrivals Service/price competition [Hassin and Haviv 2003] to incentivize strategic servers Routing Staffing

4. Outline The M/M/1 queue – a simple example Model for a strategic server The strategic M/M/N queue Classic policies in non-strategic setting Impact of strategic servers Asymptotically optimal policies Routing Staffing which idle server gets the next job? how many servers to hire?

5. M/M/1/FCFS   strategic server idleness cost utility function    LHS RHS

6. Outline The M/M/1 queue – a simple example Model for a strategic server The strategic M/M/N queue Classic policies in non-strategic setting Impact of strategic servers Asymptotically optimal policies Routing Staffing which idle server gets the next job? how many servers to hire?

7. M/M/N/FCFS  strategic servers routing   symmetric Nash equilibrium Nash equilibrium existence? performance? Blue for strategic service rates Yellow for control/policy parameters

8. Outline The M/M/1 queue – a simple example Model for a strategic server The strategic M/M/N queue Classic policies in non-strategic setting Impact of strategic servers Asymptotically optimal policies Routing Staffing which idle server gets the next job? how many servers to hire?

9. Rate-based policies Idle-time-based policies FSF SSF LISF SISF Random Common routing policies

10. M/M/N/FCFS routing    Rate-based policies Idle-time-based policies FSF SSF LISF SISF Random When servers are not strategic…

11. M/M/N/FCFS routing    Rate-based policies Idle-time-based policies FSF SSF LISF SISF Random When servers are not strategic… well-studied, asymptotically optimal [Lin and Kumar 1984] [de Véricourt et al. 2005] [Armony 2005]

12. M/M/N/FCFS routing    Rate-based policies Idle-time-based policies FSF SSF LISF SISF Random When servers are not strategic… asymptotically “fair” [Atar 2008]

13. Rate-based policies Idle-time-based policies FSF SSF LISF SISF Random When servers are strategic… M/M/N/FCFS  routing  

14. Rate-based policies Idle-time-based policies FSF SSF LISF SISF Random When servers are strategic… Our results… M/M/N/FCFS  routing  

15. Rate-based policies FSF SSF Random & Idle-time-based policies When servers are strategic… Our results… M/M/N/FCFS  routing   First order condition: same unique symmetric equilibrium

16. Rate-based policies FSF SSF Random & Idle-time-based policies When servers are strategic… Our results… M/M/N/FCFS  routing   Can we do better than Random? Yes, but…

17. Outline The M/M/1 queue – a simple example Model for a strategic server The strategic M/M/N queue Classic policies in non-strategic setting Impact of strategic servers Asymptotically optimal policies Routing Staffing which idle server gets the next job? how many servers to hire?

18. M/M/N/FCFS    When servers are not strategic… Random staffing asymptotically optimal [Borst et al. 2004]

19. M/M/N/FCFS  Random  staffing When servers are strategic… Problem: Explicit expression unknown!

20. M/M/N/FCFS  Random  staffing When servers are strategic…

21. Rate-independent Rate-dependent

22. Rate-independent Rate-dependent Limiting FOC: Policy:

23. Rate-independent Rate-dependent Limiting FOC: Policy:

24. Rate-independent Rate-dependent Limiting FOC: Policy:

25. Rate-independent Rate-dependent Limiting FOC: Policy:

26. Rate-independent Rate-dependent Limiting FOC: Policy:

27. Concluding remarks We need to rethink optimal system design when servers are strategic! Joint routing-staffing optimization? Empirical studies / Experimental evaluation? Asymmetric models / equilibria? Interaction between strategic arrivals and strategic servers? M/M/N/FCFS  Random   loss of efficiency? $ $$$$ $$ ? ?

28. Ragavendran Gopalakrishnan University of Colorado at Boulder Adam Wierman (Caltech) Amy R. Ward (USC) Sherwin Doroudi (CMU) Staffing and Routing to incentivize servers in many-server systems